I think investing in anything less than 4 cores for gaming right now is a short term gain that you will quickly regret.
Dual core gaming has pretty much reached the end of it's life right about now IMHO. With all these 4, 6, and 8 core CPUs becoming common developers can rely on most gamers having more than a dual core in future. There are already plenty of games that struggle on older i3's like the one you suggested.
Unfortunately the cheaper i5 6xxx & 7xxx series have fairly weak clock speeds, but they are still a better option.
If you could get a good deal on a 6600k or 7600k that would be Ideal. My i5 7400 games fairly well but I only play older games on it.